**Abstract**

One of the crucial issues of computational fluid dynamics is how to discretize the viscous terms accurately. Recently, an attractive and viable alternative numerical method for solving the compressible Navier–Stokes equations is proposed. The firstorder hyperbolic system (FOHS) with reconstructed discontinuous Galerkin (rDG) method was first proposed to solve advection–diffusion model equations and then extend to compressible Navier–Stokes equations. For the model advection–diffusion equation, the proposed method is reliable, accurate, efficient, and robust, benefiting from FOHS and rDG methods. To implement the method of compressible Navier– Stokes equations, the gradients of density, velocity, and temperature are introduced as auxiliary variables. Numerical experiments demonstrate that the developed HNS + rDG methods are able to achieve the designed order of accuracy for both primary variables and their gradients.

**Keywords:** reconstructed discontinuous Galerkin, hyperbolic Navier–Stokes, first-order hyperbolic system, integration methods, laminar flow
